Robust Markowitz mean-variance portfolio selection under ambiguous covariance matrix

Abstract

This paper studies a robust continuous-time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min–max mean-variance problem over a set of nondominated probability measures that is solved by a McKean–Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman–Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model.